Average Error: 10.2 → 0.0
Time: 18.7s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\mathsf{fma}\left(-\frac{x}{z}, y, y + \frac{x}{z}\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(-\frac{x}{z}, y, y + \frac{x}{z}\right)
double f(double x, double y, double z) {
        double r91329256 = x;
        double r91329257 = y;
        double r91329258 = z;
        double r91329259 = r91329258 - r91329256;
        double r91329260 = r91329257 * r91329259;
        double r91329261 = r91329256 + r91329260;
        double r91329262 = r91329261 / r91329258;
        return r91329262;
}
↓
double f(double x, double y, double z) {
        double r91329263 = x;
        double r91329264 = z;
        double r91329265 = r91329263 / r91329264;
        double r91329266 = -r91329265;
        double r91329267 = y;
        double r91329268 = r91329267 + r91329265;
        double r91329269 = fma(r91329266, r91329267, r91329268);
        return r91329269;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.2
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.2

    \[\frac{x + y \cdot \left(z - x\right)}{z}\]

  2. Simplified10.2

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}}\]

  3. Taylor expanded around 0 3.6

    \[\leadsto \color{blue}{\left(y + \frac{x}{z}\right) - \frac{x \cdot y}{z}}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(-\frac{x}{z}, y, y + \frac{x}{z}\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(-\frac{x}{z}, y, y + \frac{x}{z}\right)\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))